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We all known the problems that presents the notation of inverse/reverse/anti functions as $f^{-1}(x)$, being the most important one the confusion with ${f(x)}^{-1}$, as in the classical $\sin^{-1}(x)$, or the strange cases such as ${f^{-1}(x)}^{-2}$,etc.
There are any other notation convention for the inverse functions more or less widely used ? Something like $f^{\#}(x)$, $f^{r}(x)$, $rev\{f\}(x)$?
If not, why not ? It seems something interesting to improve.
If anything, I think $f^{-1}(x)$ is absolutely the correct notation for an inverse function. Correspondingly, I think $f^2(x)$ is absolutely the correct notation for $(f\circ f)(x) = f(f(x))$, not for $(f(x))^2$. But this is definitely a matter of taste, as well as context, and other people will disagree with me. For example, if $f$ isn't an endomorphism then $(f\circ f)$ makes no sense, whereas if the codomain of $f$ doesn't have any kind of multiplicative structure then $(f(x))^2$ doesn't make sense. As a reader, you should always keep in mind that the author may use symbols in a way more appropriate to their work (or even just their aesthetic sensibilities) and be willing to do some interpretative work!
There might be some sense in defining e.g. $f^{\circ 2}(x) = (f\circ f)(x)$, $f^{\cdot 2}(x) = (f(x))^2$, similarly to how we define $f^{\otimes 2}(x) = (f\otimes f)(x)$ on a tensor square of vector spaces $V^{\otimes 2}$, etc. I've seen this notation before. This would make the inverse function $f^{\circ(-1)}(x)$ and the reciprocal $f^{\cdot(-1)}(x)$. But this is pretty ugly, and I've almost never seen it. Unless you have a whole bunch of binary operations like $\circ, \cdot, \otimes$ lying around and you need to explicitly disambiguate between them, I'd avoid it. Just use words. (Besides which, you'd probably need to use words to tell readers what your notation means, because it's probably new to them.)
Good notation is like gold dust, but keep in mind that most notation is bad. Don't add to it, if possible. Just make your own work as readable as you can.
I was reading Tensor Geometry not too long ago and I encountered a good notation for the reverse of a map,
The reason for the two conventional uses of $^{-1}$ is that what they invert are written very similarily. In fact, in many settings, composition of two functions is actually written exactly like multiplication.
For instance, multiplying matrices corresponds to composing linear maps, so we often see composed linear maps written next to one another without a symbol between, as though we're multiplying. And when working with symmetry groups, a composition of two permutations is also often written by writing them next to one another without a symbol in-between.
So, seing as how composition of functions is often written like multiplication, it is not that difficult to see why the compositional inverse is also written the same way as the multiplicative inverse.
I would love for there to be a consistent way to differentiate between the two, but there isn't. At least not any widely used convention. Also, in some cases, like in my two examples above, it would mean you have to decide, every time you want to invert an object, whether that object is a function to be composed, or an element of a group or monoid to be multiplied, when really, you want it to be both at the same time.
Suppose we have previously established, $f:U \rightarrow V$, then unambiguous notation for the inverse can be introduced by something like "Let $g:V \rightarrow U$ be the inverse of $f$."
Standard notations already compress information extensively, sometimes to the edge of comprehensibility. Attempting further compression isn't a clear improvement to communication.
It is a cute suggestion. However, the idea of the current notation comes from a convenient shorthand $$f(f(x)) = f^2(x)$$ which allow easy "exponent" arithmetic like $$f^{-1}(f(f(x)) = f^{2-1}(x) = f(x).$$
So the current notation also has some useful features, which is the reason it was invented in the first place.
Personally, I try avoid ambiguity by distinguishing $f^{-1}(x)$ denoting the functional inverse from $f(x)^{-1}$ denoting the multiplicative inverse. In addition, for the trig functions you indicated, I use $\arcsin(x)$ instead of $\sin^{-1}(x)$, which has an additional usefulness in complex analysis since you can indicate the principal branch limitation by $\mathrm{Arcsin}(x)$, thereby distinguishing it from the general case...
This 1909 trig textbook mentions $\mathrm{inv}\sin{x}$ as an alternate form of $\arcsin{x}$ and $\sin^{-1}{x}$ and says it may be better than those because of its "general applicability". Applying this to $f(x)$ would presumably give either $\mathrm{inv\ }f(x)$ or $\mathrm{inv}f(x)$. Wikipedia also cites a 2009 textbook when it talks about putting $\mathrm{inv}$ in front of $f(x)$
I have seen $f^\leftarrow$ somewhere. I'm sorry that I can't give chapter and verse for it. It's pretty idiosyncratic, but I remember it—probably because I liked it.
In general, when there is competition between long-established notation (e.g. $f^{-1}$) and logical and intuitive, but novel, notation (e.g. $f^\leftarrow$), it is the former that wins in the long run. An exception is the present wide acceptance of $\arccos$, $\arcsin$, and so on for the inverse trigonometric and hyperbolic-trigonometric functions, in place of $\cos^{-1}$. $\sin^{-1}$, etc. However, I don't see the more general $\mathrm{arc}f$ for $f^{-1}$ catching on.
It is arguable that $f^{-1}$ is itself reasonable notation. The problem with it is that it conflicts with another convention: that of using $f^2$ (etc.) to mean $x\mapsto(f(x))^2$ rather than $x\mapsto f(f(x))$.
